Generalized Lipschitz Classes Research Articles

Titchmarsh and Boas-type theorems related to (κ,n)-Fourier transform

Abstract The aim of this paper is to prove a generalization of Titchmarsh’s theorems for the generalized Fourier transform called ( κ , n {\kappa,n} )-Fourier transform, where n is a positive integer and κ is a constant coming from Dunkl theory. As an application, we derive a ( κ , n ) {(\kappa,n)} -Fourier multiplier theorem for L 2 {L^{2}} Lipschitz spaces. Moreover, we give necessary conditions to ensure that f belongs to either one of the generalized Lipschitz classes of order m. This allows us to establish the analogue of the Boas-type result for ℱ κ , n {\mathcal{F}_{\kappa,n}} .

Boas-type results for Mellin transform

In this paper, we give necessary and sufficient conditions for a continuous on R + function f to belong the symmetric generalized Lipschitz classes defined by the Mellin translation in terms of growth or decay of some integrals connected with the Mellin transform. The results are counterparts of ones for classical Lipschitz classes and the classical (complex) Fourier transform obtained earlier by the author.

Laguerre-Bessel Transform and Generalized Lipschitz Classes

Abstract The aim of this paper is to give necessary and sufficient conditions in terms of the Fourier Laguerre-Bessel transform 𝒲 LB f of the function f to ensure that f belongs to the generalized Lipschitz classes H α k (X) and h k α (X), where X =[0, +∞) × [0, +∞).

Boas and Titchmarsh Type Theorems for Generalized Lipschitz Classes and $$q$$-Bessel Fourier Transform

Boas and Titchmarsh Type Theorems for Generalized Lipschitz Classes and $$q$$-Bessel Fourier Transform

Absolutely Convergent Fourier–Bessel Series and Generalized Lipschitz Classes

Absolutely Convergent Fourier–Bessel Series and Generalized Lipschitz Classes

Fourier-Dunkl transforms and generalized symmetric Lipschitz classes

In the paper, we give necessary and sufficient conditions for a continuous on R function f to belong the symmetric generalized Lipschitz classes defined by the Dunkl translation in terms of growth or decay of some integrals connected with the Fourier-Dunkl transform. The results are counterparts of ones for classical Lipschitz classes and the classical (complex) Fourier transform obtained earlier by the author.

Absolutely convergent Fourier–Jacobi series and generalized Lipschitz classes

In this paper, we give necessary and sufficient conditions in terms of Fourier–Jacobi coefficients of a function f, to ensure that f belongs either to one of the generalized Lipschitz classes and for and . Also a condition for generalized Jacobi differentiability of a function on interval is proved.

Approximation Properties of the Sampling Kantorovich Operators: Regularization, Saturation, Inverse Results and Favard Classes in L^p-Spaces

In the present paper, a characterization of the Favard classes for the sampling Kantorovich operators based upon bandlimited kernels has been established. In order to achieve the above result, a wide preliminary study has been necessary. First, suitable high order asymptotic type theorems in L^p-setting, 1 le p le +infty , have been proved. Then, the regularization properties of the sampling Kantorovich operators have been investigated. Here, we show how the regularity of the kernel influences the operator itself; this has been shown for bandlimited kernels, or more in general for kernels in Sobolev spaces, satisfying a Strang-Fix type condition of order r in mathbb {N}^+. Further, for the order of approximation of the sampling Kantorovich operators, quantitative estimates based on the L^p modulus of smoothness of order r have been established. As a consequence, the qualitative order of approximation is also derived assuming f in suitable Lipschitz and generalized Lipschitz classes. Moreover, an inverse theorem of approximation has been stated, allowing to obtain a full characterization of the Lipschitz and of the generalized Lipschitz classes in terms of convergence of the above sampling type series. These approximation results have been proved for not necessarily bandlimited kernels. From the above mentioned characterization, it remains uncovered the saturation case that, however, can be treated by a totally different approach assuming that the kernel is bandlimited. Indeed, since sampling Kantorovich (discrete) operators based upon bandlimited kernels can be viewed as double-singular integrals, exploiting the properties of the convolution in Fourier Analysis, we become able to get the desired result obtaining a complete overview of the approximation properties in L^p(mathbb {R}), 1 le p le +infty , for the sampling Kantorovich operators.

Error bounds of a function related to generalized Lipschitz class via the pseudo-Chebyshev wavelet and its applications in the approximation of functions

In this paper, a new computation method derived to solve the problems of approximation theory. This method is based upon pseudo-Chebyshev wavelet approximations. The pseudo-Chebyshev wavelet is being presented for the first time. The pseudo-Chebyshev wavelet is constructed by the pseudo-Chebyshev functions. The method is described and after that the error bounds of a function is analyzed. We have illustrated an example to demonstrate the accuracy and efficiency of the pseudo-Chebyshev wavelet approximation method and the main results. Four new error bounds of the function related to generalized Lipschitz class via the pseudo-Chebyshev wavelet are obtained. These estimators are the new fastest and best possible in theory of wavelet analysis.

Fourier-Bessel transforms and generalized uniform Lipschitz classes

Let , is defined on by . For f integrable on with respect to together with its Fourier-Bessel transform of order ν we give necessary and sufficient conditions to belong to the generalized Lipschitz classes and . Also a condition for generalized Bessel differentiability of a function is proved.

Approximation of functions of Lipschitz class and solution of Fokker-Planck equation by two-dimensional Legendre wavelet operational matrix

In this paper, Lipschitz class of two-variables is considered. This is the genralization of well-known Lipschitz class of functions. A new estimator of functions belonging to generalized Lipschitz class has been obtained. Also, the solutions for the Fokker-Planck equations have been obtained for two different cases by two-dimensional Legendre wavelet operational matrix method. The approximated solutions of the time-and space-Fokker Planck equation have been compared with the exact solutions and the solutions obtained by homotopy perturbation method. The proposed scheme is simple, effective and suitable for the solution of Fokker-Planck equation.

Trigonometric approximation of functions f(x,y) of generalized Lipschitz class by double Hausdorff matrix summability method

In this paper, we establish a new estimate for the degree of approximation of functions f(x,y) belonging to the generalized Lipschitz class Lip ((xi _{1}, xi _{2} );r ), r geq 1, by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from Lip ((alpha ,beta );r ) and Lip(alpha ,beta ) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and (C, gamma , delta ) means.

Approximation of a Function f Belonging to Generalized Hölder Class of Order 0 < α ≤ 1 and Generalized Lipschitz class by Haar Wavelet Method

In this paper, the approximations of a function f of generalized Holder class $$H^{(w)}_{\alpha }[0,1]$$ and generalized Lipschitz class $$Lip^{(s)}_{\alpha }[0,1]$$ have been obtained by Haar wavelet method. Calculated approximations are best possible in wavelet analysis.

Some Theorems of Approximation Theory in Weighted Smirnov Classes with Variable Exponent

Let $${ G\subset {\mathbb {C}} }$$ be a Jordan domain with rectifiable Dini smooth boundary $$\varGamma $$. In this work, we investigate approximation properties of matrix transforms constructed via Faber series in weighted Smirnov classes with variable exponent. Moreover, direct and inverse theorems of approximation theory in weighted Smirnov classes with variable exponent are proved and some results related to constructive characterization in generalized Lipschitz classes are obtained.

Wavelet Approximation of Functions Belonging to Generalized Lipschitz Class by Extended Haar Wavelet Series and its Application in the Solution of Bessel Differential Equation Using Haar Operational Matrix

In this paper, a class $$Lip_\alpha ^{(s)}[0,1]$$ is introduced. This class of functions is a generalization of known Lipschitz class, i.e., $$Lip_\alpha [0,1], 0<\alpha \le 1$$ of functions. Four new estimators $$E_N^{(1)}(f), E_{J,N}^{(2)}(f), E_N^{(3)}(f)$$ and $$E_N^{(4)}(f)$$ of functions of $$Lip_\alpha [0,1]$$ and $$Lip_\alpha ^{(s)}[0,1]$$ classes have been obtained. These estimators are sharper and best possible in the approximation of functions by wavelet methods. The developed estimators, the solution of Bessel differential equation of order zero by Haar wavelet operational matrix and its comparison with the exact solution are the main significant achievements of this research paper.

Approximation of a Function f of Generalized Lipschitz Class by Its Extended Legendre Wavelet Series

In this paper, two classes $${ Lip}_\alpha ^{(s)}[0, 1)$$ and $${ Lip}\xi [0, 1)$$ are introduced. These classes of functions are the generalization of the known Lipschitz class $${ Lip}_\alpha [0, 1), 0<\alpha \le 1$$ of functions. Four new estimators $$E_{\mu ^k,0}^{(\alpha )}(f)$$ , $$E_{\mu ^k,1}^{(\alpha )}(f)$$ , $$E_{\mu ^k,2}^{(\alpha )}(f)$$ and $$E_{\mu ^k,M}^{(\alpha )}(f)$$ of functions of $${ Lip}_\alpha ^{(s)}[0, 1)$$ class and $$E_{\mu ^k,0}^{(\xi )}(f),E_{\mu ^k,1}^{(\xi )}(f)$$ , $$E_{\mu ^k,2}^{(\xi )}(f)$$ and $$E_{\mu ^k,M}^{(\xi )}(f)$$ of functions of $${ Lip}\xi [0, 1)$$ class have been obtained. Five corollaries are deduced from the main theorems. These estimators are best possible in approximation of functions by wavelet methods.

Approximation of signals belonging to generalized Lipschitz class using -summability mean of Fourier series

Degree of approximation of functions of different classes has been studied by several researchers by different summability methods. In the proposed paper, we have established a new theorem for the approximation of a signal (function) belonging to the W(Lr,ξ(t))-class by (N¯,pn,qn)(E,s)-product summability means of a Fourier series. The result obtained here, generalizes several known theorems.

On the absolute convergence of Fourier series and generalisation of Lipschitz spaces, defined by differences of fractional order

We obtain the necessary and sufficient conditions in terms of Fourier coefficients of $2\pi$-periodic functions $f$ with absolutely convergent Fourier series, for $f$ to belong to the generalized Lipschitz classes $H^{\omega, \alpha}_{\mathbb{C}}$, and to have the fractional derivative of order $\alpha$ ($0 &lt; \alpha &lt; 1$).

Embedding of classes of functions with bounded $${\Phi}$$ Φ -variation into generalized Lipschitz spaces

In this note, sufficient and necessary conditions for embedding of classes \({\Phi}\)BV of functions with bounded \({\Phi}\)-variation in Schramm’s sense into generalized Lipschitz classes \({H_{q}^{\omega} (1 \leq q < \infty)}\) are obtained.

Embedding of generalized Lipschitz classes into classes of functions with Λ-bounded variation

In this paper, sufficient and necessary conditions for embedding of generalized Lipschitz classes Hpω, 1<p<∞ into classes ΛBV of functions with Λ-bounded variation are obtained under the mild restriction on ω.